reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem
  for P being Element of absolute ex Q being Element of absolute st P <> Q
 proof
   let P be Element of absolute;
   P in conic(1,1,-1,0,0,0);
   then P in {P where P is Point of ProjectiveSpace TOP-REAL 3:
   for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
   qfconic(1,1,-1,0,0,0,u) = 0} by PASCAL:def 2;
   then consider Q be Point of ProjectiveSpace TOP-REAL 3 such that
A1: P = Q and
A2: for u being Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
    qfconic(1,1,-1,0,0,0,u) = 0;
    consider u be Element of TOP-REAL 3 such that
A3: u is non zero and
A4: Dir u = P by ANPROJ_1:26;
A5: qfconic(1,1,-1,0,0,0,u) = 0 by A1,A2,A3,A4;
A6: u.3 <> 0 by A3,A4,Th67;
   |[u.1,u.2,-u.3]| is non zero
   proof
     assume |[u.1,u.2,-u.3]| is zero;
     then u.3 = 0 by EUCLID_5:4,FINSEQ_1:78;
     hence contradiction by A3,A4,Th67;
   end;
   then reconsider v = |[u.1,u.2,-u.3]| as non zero Element of TOP-REAL 3;
   reconsider R = Dir v as Element of ProjectiveSpace TOP-REAL 3
     by ANPROJ_1:26;
   take R;
A7: R <> P
   proof
     assume R = P;
     then are_Prop v,u by A3,A4,ANPROJ_1:22;
     then consider a be Real such that
     a <> 0 and
A8:  v = a * u by ANPROJ_1:1;
A9:  -u.3 = v`3 by EUCLID_5:2
         .= (a * u).3 by A8,EUCLID_5:def 3
         .= a * u.3 by RVSUM_1:44;
     (a + 1) * u.3 = 0 by A9;
     then a + 1 = 0 by A6; then
A10: a = -1;
A11: u.1 = v`1 by EUCLID_5:2
        .= (a * u).1 by A8,EUCLID_5:def 1
        .= a * u.1 by RVSUM_1:44;
A12: u.2 = v`2 by EUCLID_5:2
        .= (a * u).2 by A8,EUCLID_5:def 2
        .= a * u.2 by RVSUM_1:44;
     0 = 1 * u.1 * u.1 + 1 * u.2 * u.2 + (-1) * u.3 * u.3
       + 0 * u.1 * u.2 + 0 * u.1 * u.3 + 0 * u.2 * u.3 by A5,PASCAL:def 1
      .= (-1) * (u.3)^2 by A11,A10,A12;
     hence contradiction by A6;
   end;
   for w be Element of TOP-REAL 3 st w is non zero & R = Dir w holds
     qfconic(1,1,-1,0,0,0,w) = 0
   proof
     let w be Element of TOP-REAL 3;
     assume that
A13: w is non zero and
A14: R = Dir w;
     are_Prop v,w by A13,A14,ANPROJ_1:22;
     then consider b be Real such that
     b <> 0 and
A15: w = b * v by ANPROJ_1:1;
A16: w.1 = b * v.1 by A15,RVSUM_1:44
        .= b * v`1 by EUCLID_5:def 1
        .= b * u.1 by EUCLID_5:2;
A17: w.2 = b * v.2 by A15,RVSUM_1:44
        .= b * v`2 by EUCLID_5:def 2
        .= b * u.2 by EUCLID_5:2;
A18: w.3 = b * v.3 by A15,RVSUM_1:44
        .= b * v`3 by EUCLID_5:def 3
        .= b * (-u.3) by EUCLID_5:2;
     qfconic(1,1,-1,0,0,0,w) = 1 * w.1 * w.1 + 1 * w.2 * w.2
                                 + (-1) * w.3 * w.3 + 0 * w.1 * w.2
                                 + 0 * w.1 * w.3 + 0 * w.2 * w.3
                                 by PASCAL:def 1
                            .= b * b * (1 * u.1 * u.1
                                 + 1 * u.2 * u.2 + (-1) * u.3 * u.3
                                 + 0 * u.1 * u.2 + 0 * u.1 * u.3
                                 + 0 * u.2 * u.3) by A16,A17,A18
                            .= b * b * qfconic(1,1,-1,0,0,0,u) by PASCAL:def 1
                            .= 0 by A5;
     hence thesis;
   end;
   then R in {P where P is Point of ProjectiveSpace TOP-REAL 3:
     for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
     qfconic(1,1,-1,0,0,0,u) = 0};
   hence thesis by A7,PASCAL:def 2;
 end;
